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Question #40896

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(0, 4)$ and $B(4, 0)$ be two points in the $xy$ -plane. For any $t \ge 0$ , let $P(x(t), y(t))$ be the point that divides the line segment $AB$ internally in the ratio $t^2 : 1$ . Evaluate the integral $I = \int_{0}^{\sqrt{3}} t^3 \left| x(t) - y(t) \right| dt$

Options

A.
3
B.
$8 \ln 2 - 3$
C.
$3 - 8 \ln 2$
D.
$8 \ln 2 + 3$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Section Formula Internally/Externally|Substitution (Change of Variable)|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate coordinate geometry with calculus while navigating the 'Modulus Trap'.

By using the section formula, the student derives the coordinates x(t) and y(t), which form the integrand.

The critical trap occurs after the substitution u = t^2+1; the resulting integrand contains |u-2|.

Students who blindly integrate without analyzing the domain of the absolute value will fail to split the integral at u=2 (corresponding to t=1), leading to the incorrect result 8 ln(2) - 3.

Recognizing the sign change of the sectioned coordinate expression within the integration limits is essential for the correct evaluation, where the logarithmic terms beautifully cancel out.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Section Formula Internally/Externally
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free